In magnetic resonance and optical spectroscopy, it is necessary to design pulses which will selectively excite only a part of the spectrum. There are several techniques for such selective excitation, including the technique described by the present inventor in U.S. Pat. No. 5,153,515, the contents of which are hereby incorporated by reference. In that patent, the present inventor disclosed a method of constructing selective excitations such as .pi./2, .pi. and refocusing pulse sequences for perturbing the spins of a magnetic resonance imaging system. In particular, the present inventor illustrated that the desired z magnetization, M.sub.z, for a system starting at equilibrium (M.sub.z (.omega.)=1), can be written as an (N-1)th order Fourier series .omega.t, where .omega. is the off-resonance frequency. In addition, if all pulses have the same phase, then the z magnetization is symmetric in frequency (M.sub.z (.omega.)=M.sub.z (-.omega.)), and can be written as an Nth order Fourier cosine series in .omega.t. Then, the present inventor illustrated that, given a Fourier series or Fourier cosine series (in .omega.t) representing the desired z magnetization, it is possible to perform an inversion of the nonlinear problem to determine a hard pulse sequence which will actually yield the desired response. The desired z magnetization was written as a Fourier series in .omega.t using finite impulse response filter theory so that hard pulse sequences could be generated which would yield an optimal frequency response when applied to the system. It was also illustrated that a soft pulse could be generated from the hard pulse sequence which would yield the same frequency response as the hard pulse sequence.
In the direct synthesis approach described in U.S. Pat. No. 5,153,515, one specifies the desired frequency spectrum and then synthesizes an RF pulse which will yield that desired response. Unfortunately, one has little control over the shape of the RF pulse which is actually applied to the system. However, in practice, it is necessary to limit the pulse shape. For example, the peak power (instantaneous power) of the applied pulse is limited by the transmitter power. Also, the total energy used by the pulse (i.e., the integral of the instantaneous power over the pulse duration) is limited by concerns for sample or tissue heating. In practice, the total energy deposited by the RF pulse is related to the specific absorption rate (SAR), which is limited by FDA guidelines. This limitation poses severe constraints on some fast imaging schemes. As a result, it is desired to develop a technique for the synthesis of reduced power pulses with a specified frequency profile within FDA guidelines yet which also may be used to generate the desired frequency response characteristics.
As noted in U.S. Pat. No. 5,153,515, the relationship between a radio frequency pulse and the frequency response of its effects on even a simple spin system is complex and nonlinear. The nonlinearity of this relationship has led to difficulties in the analysis and inversion of this relationship. However, over the last several years, many of these difficulties have been overcome. For example, given a desired frequency profile, the present inventor has proposed in U.S. Pat. No. 5,153,515 an algorithm which allows for the generation of pulses that will yield the desired response. Other systems for generating a desired frequency profile have also been proposed by Pauly et al. in an article entitled "Parameter Relations for the Shinnar-Le Roux Selective Excitation Pulse Design Algorithm," IEEE Trans. Med. Imag., Vol. 10, No. 1, pp. 53-65 (1991); Carlson in an article entitled "Exact Solutions for Selective-Excitation Pulses," J. Magn. Reson., Vol. 94, pp. 376-386 (1991); and Yagle in an article entitled "Inversion of the Bloch Transform in Magnetic Resonance Imaging Using Asymmetric Two-Component Inverse Scattering," Inverse Problems, Vol. 6, pp. 133-151 (1990). However, such algorithms offer only a partial solution in that there is still no illustration of the relationship between the details of the pulse shape and the frequency response. For example, if the pulse is restricted so that either the total energy or the peak power is bounded by a preset maximum, the limits this places on the possible frequency response have heretofore been unknown. Furthermore, outside of using a search algorithm such as that described by Conolly et al. in an article entitled "Optimal Control Solutions to the Magnetic Resonance Selective Excitation Problem," IEEE Trans. Med. Imaging, Vol. 5, No. 2, pp. 106-115 (1986), there is no known algorithm for incorporating constraints on the energy into the direct synthesis algorithms.
Recently, the present inventor obtained a partial solution to the relationship between a pulse and its frequency response. In particular, it was illustrated by Shinnar et al. in an article entitled "Inversion of the Bloch Equation," J. Chem. Phys., Vol. 98, pp. 6121-6128 (1993), that if pulses are restricted to be of finite time duration T, then the Fourier transform of the frequency response function is nonzero only over a finite period in the time domain. In other words, if the z magnetization, as a function of frequency .omega., is Fourier transformed back into the time domain s, it is zero for all .linevert split.s.linevert split.&gt;T. Similar results have been obtained for all other functions which can be used to describe the frequency response. The present inventor further showed that one could not have a frequency response that had a sharper transition zone between excited and unexcited frequencies than the Fourier transform limit, 1/T. This solved the open problem about the limits achievable by RF pulses described by McDonald et al. in an article entitled "Testing the Limits of Shape Optimization by Large-Flip-Angle Pulses," J. Magn. Reson., Vol. 99, pp. 282-291 (1992).
Thus, if pulses are restricted to a finite time duration T, the Fourier transform of the frequency response of the RF pulse completely characterizes the space of magnetizations reachable with a finite duration RF pulse. In other words, not only must any magnetization profile achievable by a finite duration RF pulse satisfy this constraint, but also, for any magnetization function which satisfies this constraint, a finite duration pulse can be synthesized which yields the desired magnetization.
In practice, many other limitations on the pulse shape are useful. For example, the total energy and peak power used by a pulse are clearly of major significance because of sample heating and hardware limitations. As noted above, in clinical imaging, specific absorption rate (SAR) significantly limits some fast imaging schemes. Unfortunately, to date, there exists no theory or method known to the inventor for directly relating the energy of a pulse to the frequency response function. At present, one has to invert the Bloch equation and then calculate the energy. To the inventor's knowledge, no one has suggested calculating the energy requirement directly from the frequency response function.
It is thus desired to develop a technique which relates the energy requirement to the frequency response function so that the peak power of the synthesized "optimal" RF pulses can be reduced without any change in the excitation profile, time of the pulse, or the total energy of the pulse. Also, it is desired to develop a technique in which the total energy of the pulse is reduced as well. The method of the present invention has been developed for these purposes.